SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1097-A1115, January 2020.
This paper introduces a “kernel-independent” interpolative decomposition butterfly factorization (IDBF) as a data-sparse approximation for matrices that satisfy a complementary low-rank property. The IDBF can be constructed in $O(N\log N)$ operations for an $N\times N$ matrix via hierarchical interpolative decompositions (IDs) if matrix entries can be sampled individually and each sample takes $O(1)$ operations. The resulting factorization is a product of $O(\log N)$ sparse matrices, each with $O(N)$ nonzero entries. Hence, it can be applied to a vector rapidly in $O(N\log N)$ operations. IDBF is a general framework for nearly optimal fast matrix-vector multiplication (matvec), which is useful in a wide range of applications, e.g., special function transformation, Fourier integral operators, and high-frequency wave computation. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.

中文翻译：

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插值分解蝶式分解

SIAM科学计算杂志，第42卷，第2期，第A1097-A1115页，2020年1月。
本文介绍了一种“核无关”插值分解蝶形分解（IDBF），它是满足互补低秩属性的矩阵的数据稀疏近似。如果矩阵项可以单独采样并且每个样本执行$ O（1）$运算，则可以通过层次插值分解（ID）对$ N \ times N $矩阵以$ O（N \ log N）$运算构造IDBF。 。所得的因式分解是$ O（\ log N）$个稀疏矩阵的乘积，每个矩阵都有$ O（N）$个非零条目。因此，它可以在$ O（N \ log N）$运算中快速应用于向量。IDBF是用于近乎最佳的快速矩阵矢量乘法（matvec）的通用框架，可用于各种应用程序，例如特殊函数变换，傅立叶积分算子和高频波计算。